1547edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo| 2452 -1547 }} | | {{monzo| 2452 -1547 }} | ||
| {{mapping| 1547 2542 }} | | {{mapping| 1547 2542 }} | ||
| | | −0.015 | ||
| 0.015 | | 0.015 | ||
| 1.99 | | 1.99 | ||
| Line 38: | Line 30: | ||
| {{monzo| -52 -17 34 }}, {{monzo| 40 -56 21 }} | | {{monzo| -52 -17 34 }}, {{monzo| 40 -56 21 }} | ||
| {{mapping| 1547 2542 3592 }} | | {{mapping| 1547 2542 3592 }} | ||
| | | −0.008 | ||
| 0.017 | | 0.017 | ||
| 2.14 | | 2.14 | ||
| Line 45: | Line 37: | ||
| 4375/4374, {{monzo| -1 4 11 -11 }}, {{monzo| 46 -14 -3 -6 }} | | 4375/4374, {{monzo| -1 4 11 -11 }}, {{monzo| 46 -14 -3 -6 }} | ||
| {{mapping| 1547 2542 3592 4343 }} | | {{mapping| 1547 2542 3592 4343 }} | ||
| | | −0.007 | ||
| 0.014 | | 0.014 | ||
| 1.86 | | 1.86 | ||
| Line 52: | Line 44: | ||
| 4375/4374, 117649/117612, 234375/234256, 2097152/2096325 | | 4375/4374, 117649/117612, 234375/234256, 2097152/2096325 | ||
| {{mapping| 1547 2542 3592 4343 5352 }} | | {{mapping| 1547 2542 3592 4343 5352 }} | ||
| | | −0.017 | ||
| 0.024 | | 0.024 | ||
| 3.10 | | 3.10 | ||
| Line 59: | Line 51: | ||
| 4096/4095, 4375/4374, 6656/6655, 78125/78078, 85750/85683 | | 4096/4095, 4375/4374, 6656/6655, 78125/78078, 85750/85683 | ||
| {{mapping| 1547 2542 3592 4343 5352 5725 }} | | {{mapping| 1547 2542 3592 4343 5352 5725 }} | ||
| | | −0.029 | ||
| 0.034 | | 0.034 | ||
| 4.42 | | 4.42 | ||
{{comma basis end}} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 87: | Line 72: | ||
|- | |- | ||
| 7 | | 7 | ||
| 670\1547<br>(7\1547) | | 670\1547<br />(7\1547) | ||
| 519.715<br>(5.429) | | 519.715<br />(5.429) | ||
| 27/20<br>(325/324) | | 27/20<br />(325/324) | ||
| [[Brahmagupta]] (7-limit) | | [[Brahmagupta]] (7-limit) | ||
|- | |- | ||
| Line 99: | Line 84: | ||
|- | |- | ||
| 13 | | 13 | ||
| 642\1547<br>(47\1547) | | 642\1547<br />(47\1547) | ||
| 497.996<br>(36.458) | | 497.996<br />(36.458) | ||
| 4/3<br>(?) | | 4/3<br />(?) | ||
| [[Aluminium]] | | [[Aluminium]] | ||
|- | |- | ||
| 17 | | 17 | ||
| 321\1547<br>(48\1547) | | 321\1547<br />(48\1547) | ||
| 248.998<br>(37.233) | | 248.998<br />(37.233) | ||
| {{monzo| -23 5 9 -2 }}<br>(100352/98415) | | {{monzo| -23 5 9 -2 }}<br />(100352/98415) | ||
| [[Chlorine]] | | [[Chlorine]] | ||
|- | |- | ||
| 91 | | 91 | ||
| 642\1547<br>(13\1547) | | 642\1547<br />(13\1547) | ||
| 497.996<br>(10.084) | | 497.996<br />(10.084) | ||
| 4/3<br>(176/175) | | 4/3<br />(176/175) | ||
| [[Protactinium]] | | [[Protactinium]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
== Music == | == Music == | ||
Revision as of 04:38, 16 November 2024
| ← 1546edo | 1547edo | 1548edo → |
Theory
1547edo is consistent to the 15-odd-limit and is excellent in the 7-limit. It tempers out 4375/4374 and it is a member of the optimal ET sequence for the rank-3 temperament associated with this comma.
In the 5-limit, it supports gross, which is a very high-accuracy temperament. The 118-tone maximal evenness scale produced by gross is concoctic, since it uses 118\1547 as the generator. In addition, 1547edo tempers out the septendecima and thus supports the chlorine temperament in 5-limit and also in the 7-limit. 1547edo tempers out the 5-limit comma [236 -61 -60⟩, thus associating a stack of sixty 15/8's with 4/3, and sixty-one of them make 5/4.
In the 7-limit, it provides the optimal patent val for 7-limit brahmagupta, the 441 & 1106 temperament, and supports an alternative 11-limit extension to it. It also supports semidimi, the 171 & 1376 temperament.
In the 11-limit, 1547edo provides the optimal patent val for the aluminium temperament, which maps 135/128 to 1/13th of the occtave. It also tempers out 117649/117612, and is a tuning for the rank-3 temperament heimdall. In higher limits, it supports 91th-octave temperament protactinium.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.049 | -0.018 | +0.017 | +0.201 | +0.326 | -0.237 | +0.354 | +0.039 | -0.230 | -0.110 |
| Relative (%) | +0.0 | +6.3 | -2.3 | +2.2 | +25.9 | +42.0 | -30.5 | +45.6 | +5.0 | -29.7 | -14.2 | |
| Steps (reduced) |
1547 (0) |
2452 (905) |
3592 (498) |
4343 (1249) |
5352 (711) |
5725 (1084) |
6323 (135) |
6572 (384) |
6998 (810) |
7515 (1327) |
7664 (1476) | |
Subsets and supersets
Since 1547 factors into 7 × 13 × 17, 1547edo has subset edos 7, 13, 17, 91, 119, and 221.
Regular temperament properties
Template:Comma basis begin |- | 2.3 | [2452 -1547⟩ | [⟨1547 2542]] | −0.015 | 0.015 | 1.99 |- | 2.3.5 | [-52 -17 34⟩, [40 -56 21⟩ | [⟨1547 2542 3592]] | −0.008 | 0.017 | 2.14 |- | 2.3.5.7 | 4375/4374, [-1 4 11 -11⟩, [46 -14 -3 -6⟩ | [⟨1547 2542 3592 4343]] | −0.007 | 0.014 | 1.86 |- | 2.3.5.7.11 | 4375/4374, 117649/117612, 234375/234256, 2097152/2096325 | [⟨1547 2542 3592 4343 5352]] | −0.017 | 0.024 | 3.10 |- | 2.3.5.7.11.13 | 4096/4095, 4375/4374, 6656/6655, 78125/78078, 85750/85683 | [⟨1547 2542 3592 4343 5352 5725]] | −0.029 | 0.034 | 4.42 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 1
| 118\1547
| 91.532
| [9 -32 18⟩
| Gross
|-
| 1
| 579\1547
| 449.127
| 35/27
| Semidimi
|-
| 7
| 670\1547
(7\1547)
| 519.715
(5.429)
| 27/20
(325/324)
| Brahmagupta (7-limit)
|-
| 7
| 11\1547
| 8.533
| 1029/1024
| Nitrogen
|-
| 13
| 642\1547
(47\1547)
| 497.996
(36.458)
| 4/3
(?)
| Aluminium
|-
| 17
| 321\1547
(48\1547)
| 248.998
(37.233)
| [-23 5 9 -2⟩
(100352/98415)
| Chlorine
|-
| 91
| 642\1547
(13\1547)
| 497.996
(10.084)
| 4/3
(176/175)
| Protactinium
Template:Rank-2 end
Template:Orf